nonlocal evolution equations, sqg · collaborators i tarek elgindi (princeton) i mihaela ignatova,...
TRANSCRIPT
![Page 1: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/1.jpg)
Nonlocal evolution equations, SQG
Peter Constantin
IMA, June 2016
![Page 2: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/2.jpg)
Collaborators
I Tarek Elgindi (Princeton)I Mihaela Ignatova, (Princeton)I Vlad Vicol (Princeton)
![Page 3: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/3.jpg)
Nonlocality in fluids:
I incompressibility: pressureI irrotationality: Dirichlet-to Neumann map in water wavesI anisotropy: dissipative fractional Laplacians: sqg, mhdI confined matter, unconfined fields: electroconvection
![Page 4: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/4.jpg)
Nonlocality in fluids:I incompressibility: pressure
I irrotationality: Dirichlet-to Neumann map in water wavesI anisotropy: dissipative fractional Laplacians: sqg, mhdI confined matter, unconfined fields: electroconvection
![Page 5: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/5.jpg)
Nonlocality in fluids:I incompressibility: pressureI irrotationality: Dirichlet-to Neumann map in water waves
I anisotropy: dissipative fractional Laplacians: sqg, mhdI confined matter, unconfined fields: electroconvection
![Page 6: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/6.jpg)
Nonlocality in fluids:I incompressibility: pressureI irrotationality: Dirichlet-to Neumann map in water wavesI anisotropy: dissipative fractional Laplacians: sqg, mhd
I confined matter, unconfined fields: electroconvection
![Page 7: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/7.jpg)
Nonlocality in fluids:I incompressibility: pressureI irrotationality: Dirichlet-to Neumann map in water wavesI anisotropy: dissipative fractional Laplacians: sqg, mhdI confined matter, unconfined fields: electroconvection
![Page 8: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/8.jpg)
SQG without boundaries
∂tθ + u · ∇θ = 0
Active scalar:u = R⊥θ
in Rd or Td .R = ∇(−∆)−
12
R⊥ = MR with M invertible, antisymmetric. In d = 2, rotation by 90degrees. Makes u divergence-free. In Fourier
R⊥θ(k) = ik⊥
|k |θ(k)
SQG– geophysical origin: Charney. Held, SwansonC-Majda-Tabak: analogies to 3D Euler.
![Page 9: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/9.jpg)
SQG without boundaries
∂tθ + u · ∇θ = 0
Active scalar:
u = R⊥θ
in Rd or Td .R = ∇(−∆)−
12
R⊥ = MR with M invertible, antisymmetric. In d = 2, rotation by 90degrees. Makes u divergence-free. In Fourier
R⊥θ(k) = ik⊥
|k |θ(k)
SQG– geophysical origin: Charney. Held, SwansonC-Majda-Tabak: analogies to 3D Euler.
![Page 10: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/10.jpg)
SQG without boundaries
∂tθ + u · ∇θ = 0
Active scalar:u = R⊥θ
in Rd or Td .R = ∇(−∆)−
12
R⊥ = MR with M invertible, antisymmetric. In d = 2, rotation by 90degrees. Makes u divergence-free. In Fourier
R⊥θ(k) = ik⊥
|k |θ(k)
SQG– geophysical origin: Charney. Held, SwansonC-Majda-Tabak: analogies to 3D Euler.
![Page 11: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/11.jpg)
SQG without boundaries
∂tθ + u · ∇θ = 0
Active scalar:u = R⊥θ
in Rd or Td .
R = ∇(−∆)−12
R⊥ = MR with M invertible, antisymmetric. In d = 2, rotation by 90degrees. Makes u divergence-free. In Fourier
R⊥θ(k) = ik⊥
|k |θ(k)
SQG– geophysical origin: Charney. Held, SwansonC-Majda-Tabak: analogies to 3D Euler.
![Page 12: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/12.jpg)
SQG without boundaries
∂tθ + u · ∇θ = 0
Active scalar:u = R⊥θ
in Rd or Td .R = ∇(−∆)−
12
R⊥ = MR with M invertible, antisymmetric.
In d = 2, rotation by 90degrees. Makes u divergence-free. In Fourier
R⊥θ(k) = ik⊥
|k |θ(k)
SQG– geophysical origin: Charney. Held, SwansonC-Majda-Tabak: analogies to 3D Euler.
![Page 13: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/13.jpg)
SQG without boundaries
∂tθ + u · ∇θ = 0
Active scalar:u = R⊥θ
in Rd or Td .R = ∇(−∆)−
12
R⊥ = MR with M invertible, antisymmetric. In d = 2, rotation by 90degrees.
Makes u divergence-free. In Fourier
R⊥θ(k) = ik⊥
|k |θ(k)
SQG– geophysical origin: Charney. Held, SwansonC-Majda-Tabak: analogies to 3D Euler.
![Page 14: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/14.jpg)
SQG without boundaries
∂tθ + u · ∇θ = 0
Active scalar:u = R⊥θ
in Rd or Td .R = ∇(−∆)−
12
R⊥ = MR with M invertible, antisymmetric. In d = 2, rotation by 90degrees. Makes u divergence-free.
In Fourier
R⊥θ(k) = ik⊥
|k |θ(k)
SQG– geophysical origin: Charney. Held, SwansonC-Majda-Tabak: analogies to 3D Euler.
![Page 15: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/15.jpg)
SQG without boundaries
∂tθ + u · ∇θ = 0
Active scalar:u = R⊥θ
in Rd or Td .R = ∇(−∆)−
12
R⊥ = MR with M invertible, antisymmetric. In d = 2, rotation by 90degrees. Makes u divergence-free. In Fourier
R⊥θ(k) = ik⊥
|k |θ(k)
SQG– geophysical origin: Charney. Held, SwansonC-Majda-Tabak: analogies to 3D Euler.
![Page 16: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/16.jpg)
SQG without boundaries
∂tθ + u · ∇θ = 0
Active scalar:u = R⊥θ
in Rd or Td .R = ∇(−∆)−
12
R⊥ = MR with M invertible, antisymmetric. In d = 2, rotation by 90degrees. Makes u divergence-free. In Fourier
R⊥θ(k) = ik⊥
|k |θ(k)
SQG– geophysical origin: Charney.
Held, SwansonC-Majda-Tabak: analogies to 3D Euler.
![Page 17: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/17.jpg)
SQG without boundaries
∂tθ + u · ∇θ = 0
Active scalar:u = R⊥θ
in Rd or Td .R = ∇(−∆)−
12
R⊥ = MR with M invertible, antisymmetric. In d = 2, rotation by 90degrees. Makes u divergence-free. In Fourier
R⊥θ(k) = ik⊥
|k |θ(k)
SQG– geophysical origin: Charney. Held, Swanson
C-Majda-Tabak: analogies to 3D Euler.
![Page 18: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/18.jpg)
SQG without boundaries
∂tθ + u · ∇θ = 0
Active scalar:u = R⊥θ
in Rd or Td .R = ∇(−∆)−
12
R⊥ = MR with M invertible, antisymmetric. In d = 2, rotation by 90degrees. Makes u divergence-free. In Fourier
R⊥θ(k) = ik⊥
|k |θ(k)
SQG– geophysical origin: Charney. Held, SwansonC-Majda-Tabak: analogies to 3D Euler.
![Page 19: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/19.jpg)
Motivation: 3D Euler analogy
∇⊥θ like 3D vorticity.
Levels of theta = lines frozen in the flow:
[(∂t + u · ∇), (∇⊥θ · ∇
)] = 0
Kinetic energy conserved:
ddt
|u(x , t)|2dx = 0
∂t (∇⊥θ) + u · ∇(∇⊥θ) = (∇u)(∇⊥θ)
Stretching term like 3D Euler, 6= 0. Blow-up problem, open. Directionof level lines locally nice⇒ depletion of nonlinearity.
![Page 20: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/20.jpg)
Motivation: 3D Euler analogy
∇⊥θ like 3D vorticity. Levels of theta = lines frozen in the flow:
[(∂t + u · ∇), (∇⊥θ · ∇
)] = 0
Kinetic energy conserved:
ddt
|u(x , t)|2dx = 0
∂t (∇⊥θ) + u · ∇(∇⊥θ) = (∇u)(∇⊥θ)
Stretching term like 3D Euler, 6= 0. Blow-up problem, open. Directionof level lines locally nice⇒ depletion of nonlinearity.
![Page 21: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/21.jpg)
Motivation: 3D Euler analogy
∇⊥θ like 3D vorticity. Levels of theta = lines frozen in the flow:
[(∂t + u · ∇), (∇⊥θ · ∇
)] = 0
Kinetic energy conserved:
ddt
|u(x , t)|2dx = 0
∂t (∇⊥θ) + u · ∇(∇⊥θ) = (∇u)(∇⊥θ)
Stretching term like 3D Euler, 6= 0. Blow-up problem, open. Directionof level lines locally nice⇒ depletion of nonlinearity.
![Page 22: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/22.jpg)
Motivation: 3D Euler analogy
∇⊥θ like 3D vorticity. Levels of theta = lines frozen in the flow:
[(∂t + u · ∇), (∇⊥θ · ∇
)] = 0
Kinetic energy conserved:
ddt
|u(x , t)|2dx = 0
∂t (∇⊥θ) + u · ∇(∇⊥θ) = (∇u)(∇⊥θ)
Stretching term like 3D Euler, 6= 0. Blow-up problem, open. Directionof level lines locally nice⇒ depletion of nonlinearity.
![Page 23: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/23.jpg)
Motivation: 3D Euler analogy
∇⊥θ like 3D vorticity. Levels of theta = lines frozen in the flow:
[(∂t + u · ∇), (∇⊥θ · ∇
)] = 0
Kinetic energy conserved:
ddt
|u(x , t)|2dx = 0
∂t (∇⊥θ) + u · ∇(∇⊥θ) = (∇u)(∇⊥θ)
Stretching term like 3D Euler, 6= 0. Blow-up problem, open.
Directionof level lines locally nice⇒ depletion of nonlinearity.
![Page 24: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/24.jpg)
Motivation: 3D Euler analogy
∇⊥θ like 3D vorticity. Levels of theta = lines frozen in the flow:
[(∂t + u · ∇), (∇⊥θ · ∇
)] = 0
Kinetic energy conserved:
ddt
|u(x , t)|2dx = 0
∂t (∇⊥θ) + u · ∇(∇⊥θ) = (∇u)(∇⊥θ)
Stretching term like 3D Euler, 6= 0. Blow-up problem, open. Directionof level lines locally nice
⇒ depletion of nonlinearity.
![Page 25: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/25.jpg)
Motivation: 3D Euler analogy
∇⊥θ like 3D vorticity. Levels of theta = lines frozen in the flow:
[(∂t + u · ∇), (∇⊥θ · ∇
)] = 0
Kinetic energy conserved:
ddt
|u(x , t)|2dx = 0
∂t (∇⊥θ) + u · ∇(∇⊥θ) = (∇u)(∇⊥θ)
Stretching term like 3D Euler, 6= 0. Blow-up problem, open. Directionof level lines locally nice⇒ depletion of nonlinearity.
![Page 26: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/26.jpg)
Critical (dissipative) SQG, Rd
∂tθ + u · ∇θ + Λθ = 0,u = R⊥θ
Λ = (−∆)12 , R = ∇Λ−1
In Fourier:Λθ(k) = |k |θ(k), Rθ(k) =
ik|k |θ(k).
I transport + nonlocal diffusion ⇒ L∞
I L∞ not good for CZ operatorsI quasilinear, critical: no room
![Page 27: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/27.jpg)
Critical (dissipative) SQG, Rd
∂tθ + u · ∇θ + Λθ = 0,u = R⊥θ
Λ = (−∆)12 , R = ∇Λ−1
In Fourier:Λθ(k) = |k |θ(k), Rθ(k) =
ik|k |θ(k).
I transport + nonlocal diffusion ⇒ L∞
I L∞ not good for CZ operatorsI quasilinear, critical: no room
![Page 28: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/28.jpg)
Critical (dissipative) SQG, Rd
∂tθ + u · ∇θ + Λθ = 0,u = R⊥θ
Λ = (−∆)12 , R = ∇Λ−1
In Fourier:Λθ(k) = |k |θ(k), Rθ(k) =
ik|k |θ(k).
I transport + nonlocal diffusion ⇒ L∞
I L∞ not good for CZ operatorsI quasilinear, critical: no room
![Page 29: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/29.jpg)
Critical (dissipative) SQG, Rd
∂tθ + u · ∇θ + Λθ = 0,u = R⊥θ
Λ = (−∆)12 , R = ∇Λ−1
In Fourier:Λθ(k) = |k |θ(k), Rθ(k) =
ik|k |θ(k).
I transport + nonlocal diffusion ⇒ L∞
I L∞ not good for CZ operatorsI quasilinear, critical: no room
![Page 30: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/30.jpg)
Critical (dissipative) SQG, Rd
∂tθ + u · ∇θ + Λθ = 0,u = R⊥θ
Λ = (−∆)12 , R = ∇Λ−1
In Fourier:Λθ(k) = |k |θ(k), Rθ(k) =
ik|k |θ(k).
I transport + nonlocal diffusion ⇒ L∞
I L∞ not good for CZ operators
I quasilinear, critical: no room
![Page 31: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/31.jpg)
Critical (dissipative) SQG, Rd
∂tθ + u · ∇θ + Λθ = 0,u = R⊥θ
Λ = (−∆)12 , R = ∇Λ−1
In Fourier:Λθ(k) = |k |θ(k), Rθ(k) =
ik|k |θ(k).
I transport + nonlocal diffusion ⇒ L∞
I L∞ not good for CZ operatorsI quasilinear, critical: no room
![Page 32: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/32.jpg)
Regularity and Uniqueness
Regularity and uniqueness: with critical dissipation: Cordoba-Wu-C =small data in L∞.
Large data: many methods (by now):1. Kiselev-Nazarov-Volberg: Maximum priciple for a modulus ofcontinuity. adequate h(r) so that
|θ0(x)− θ0(y)| < h(|x − y |)⇒ |θ(x , t)− θ(y , t)| < h(|x − y |)
2. Caffarelli-Vasseur: de Giorgi strategy: from L2 to L∞, from L∞ toCα, from Cα to C∞.3. Kiselev-Nazarov: duality method, co-evolving molecules.4. C-Vicol: nonlinear maximum principle, stability of the “only smallshocks” condition5. C-Tarfulea-Vicol: nonlinear maximum principle, small Holderexponent.
![Page 33: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/33.jpg)
Regularity and Uniqueness
Regularity and uniqueness: with critical dissipation: Cordoba-Wu-C =small data in L∞. Large data: many methods (by now):
1. Kiselev-Nazarov-Volberg: Maximum priciple for a modulus ofcontinuity. adequate h(r) so that
|θ0(x)− θ0(y)| < h(|x − y |)⇒ |θ(x , t)− θ(y , t)| < h(|x − y |)
2. Caffarelli-Vasseur: de Giorgi strategy: from L2 to L∞, from L∞ toCα, from Cα to C∞.3. Kiselev-Nazarov: duality method, co-evolving molecules.4. C-Vicol: nonlinear maximum principle, stability of the “only smallshocks” condition5. C-Tarfulea-Vicol: nonlinear maximum principle, small Holderexponent.
![Page 34: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/34.jpg)
Regularity and Uniqueness
Regularity and uniqueness: with critical dissipation: Cordoba-Wu-C =small data in L∞. Large data: many methods (by now):1. Kiselev-Nazarov-Volberg: Maximum priciple for a modulus ofcontinuity.
adequate h(r) so that
|θ0(x)− θ0(y)| < h(|x − y |)⇒ |θ(x , t)− θ(y , t)| < h(|x − y |)
2. Caffarelli-Vasseur: de Giorgi strategy: from L2 to L∞, from L∞ toCα, from Cα to C∞.3. Kiselev-Nazarov: duality method, co-evolving molecules.4. C-Vicol: nonlinear maximum principle, stability of the “only smallshocks” condition5. C-Tarfulea-Vicol: nonlinear maximum principle, small Holderexponent.
![Page 35: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/35.jpg)
Regularity and Uniqueness
Regularity and uniqueness: with critical dissipation: Cordoba-Wu-C =small data in L∞. Large data: many methods (by now):1. Kiselev-Nazarov-Volberg: Maximum priciple for a modulus ofcontinuity. adequate h(r) so that
|θ0(x)− θ0(y)| < h(|x − y |)⇒ |θ(x , t)− θ(y , t)| < h(|x − y |)
2. Caffarelli-Vasseur: de Giorgi strategy: from L2 to L∞, from L∞ toCα, from Cα to C∞.3. Kiselev-Nazarov: duality method, co-evolving molecules.4. C-Vicol: nonlinear maximum principle, stability of the “only smallshocks” condition5. C-Tarfulea-Vicol: nonlinear maximum principle, small Holderexponent.
![Page 36: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/36.jpg)
Regularity and Uniqueness
Regularity and uniqueness: with critical dissipation: Cordoba-Wu-C =small data in L∞. Large data: many methods (by now):1. Kiselev-Nazarov-Volberg: Maximum priciple for a modulus ofcontinuity. adequate h(r) so that
|θ0(x)− θ0(y)| < h(|x − y |)⇒ |θ(x , t)− θ(y , t)| < h(|x − y |)
2. Caffarelli-Vasseur: de Giorgi strategy:
from L2 to L∞, from L∞ toCα, from Cα to C∞.3. Kiselev-Nazarov: duality method, co-evolving molecules.4. C-Vicol: nonlinear maximum principle, stability of the “only smallshocks” condition5. C-Tarfulea-Vicol: nonlinear maximum principle, small Holderexponent.
![Page 37: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/37.jpg)
Regularity and Uniqueness
Regularity and uniqueness: with critical dissipation: Cordoba-Wu-C =small data in L∞. Large data: many methods (by now):1. Kiselev-Nazarov-Volberg: Maximum priciple for a modulus ofcontinuity. adequate h(r) so that
|θ0(x)− θ0(y)| < h(|x − y |)⇒ |θ(x , t)− θ(y , t)| < h(|x − y |)
2. Caffarelli-Vasseur: de Giorgi strategy: from L2 to L∞, from L∞ toCα, from Cα to C∞.
3. Kiselev-Nazarov: duality method, co-evolving molecules.4. C-Vicol: nonlinear maximum principle, stability of the “only smallshocks” condition5. C-Tarfulea-Vicol: nonlinear maximum principle, small Holderexponent.
![Page 38: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/38.jpg)
Regularity and Uniqueness
Regularity and uniqueness: with critical dissipation: Cordoba-Wu-C =small data in L∞. Large data: many methods (by now):1. Kiselev-Nazarov-Volberg: Maximum priciple for a modulus ofcontinuity. adequate h(r) so that
|θ0(x)− θ0(y)| < h(|x − y |)⇒ |θ(x , t)− θ(y , t)| < h(|x − y |)
2. Caffarelli-Vasseur: de Giorgi strategy: from L2 to L∞, from L∞ toCα, from Cα to C∞.3. Kiselev-Nazarov: duality method, co-evolving molecules.
4. C-Vicol: nonlinear maximum principle, stability of the “only smallshocks” condition5. C-Tarfulea-Vicol: nonlinear maximum principle, small Holderexponent.
![Page 39: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/39.jpg)
Regularity and Uniqueness
Regularity and uniqueness: with critical dissipation: Cordoba-Wu-C =small data in L∞. Large data: many methods (by now):1. Kiselev-Nazarov-Volberg: Maximum priciple for a modulus ofcontinuity. adequate h(r) so that
|θ0(x)− θ0(y)| < h(|x − y |)⇒ |θ(x , t)− θ(y , t)| < h(|x − y |)
2. Caffarelli-Vasseur: de Giorgi strategy: from L2 to L∞, from L∞ toCα, from Cα to C∞.3. Kiselev-Nazarov: duality method, co-evolving molecules.4. C-Vicol: nonlinear maximum principle, stability of the “only smallshocks” condition
5. C-Tarfulea-Vicol: nonlinear maximum principle, small Holderexponent.
![Page 40: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/40.jpg)
Regularity and Uniqueness
Regularity and uniqueness: with critical dissipation: Cordoba-Wu-C =small data in L∞. Large data: many methods (by now):1. Kiselev-Nazarov-Volberg: Maximum priciple for a modulus ofcontinuity. adequate h(r) so that
|θ0(x)− θ0(y)| < h(|x − y |)⇒ |θ(x , t)− θ(y , t)| < h(|x − y |)
2. Caffarelli-Vasseur: de Giorgi strategy: from L2 to L∞, from L∞ toCα, from Cα to C∞.3. Kiselev-Nazarov: duality method, co-evolving molecules.4. C-Vicol: nonlinear maximum principle, stability of the “only smallshocks” condition5. C-Tarfulea-Vicol: nonlinear maximum principle, small Holderexponent.
![Page 41: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/41.jpg)
Forced Critical SQG
SQG+f
∂tθ + (R⊥θ) · ∇θ + Λθ = f
in T2, with θ0 ∈ H1,
H1(T2) = θ |
T2θdx = 0,
T2|∇θ|2dx <∞
f ∈ H1 ∩ L∞. Let S(t)θ0 denote the solution.
Theorem(C, Tarfulea, Vicol, ’13). ∃!X ⊂ H1,
limt→∞
distH1 (S(t)θ0,X ) = 0,
uniform for θ0 in bounded sets, X invariant S(t)X = X, compact, asnice (C∞) as forces permit, and dF (X ) <∞.
![Page 42: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/42.jpg)
Forced Critical SQG
SQG+f
∂tθ + (R⊥θ) · ∇θ + Λθ = f
in T2,
with θ0 ∈ H1,
H1(T2) = θ |
T2θdx = 0,
T2|∇θ|2dx <∞
f ∈ H1 ∩ L∞. Let S(t)θ0 denote the solution.
Theorem(C, Tarfulea, Vicol, ’13). ∃!X ⊂ H1,
limt→∞
distH1 (S(t)θ0,X ) = 0,
uniform for θ0 in bounded sets, X invariant S(t)X = X, compact, asnice (C∞) as forces permit, and dF (X ) <∞.
![Page 43: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/43.jpg)
Forced Critical SQG
SQG+f
∂tθ + (R⊥θ) · ∇θ + Λθ = f
in T2, with θ0 ∈ H1,
H1(T2) = θ |
T2θdx = 0,
T2|∇θ|2dx <∞
f ∈ H1 ∩ L∞. Let S(t)θ0 denote the solution.
Theorem(C, Tarfulea, Vicol, ’13). ∃!X ⊂ H1,
limt→∞
distH1 (S(t)θ0,X ) = 0,
uniform for θ0 in bounded sets, X invariant S(t)X = X, compact, asnice (C∞) as forces permit, and dF (X ) <∞.
![Page 44: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/44.jpg)
Forced Critical SQG
SQG+f
∂tθ + (R⊥θ) · ∇θ + Λθ = f
in T2, with θ0 ∈ H1,
H1(T2) = θ |
T2θdx = 0,
T2|∇θ|2dx <∞
f ∈ H1 ∩ L∞.
Let S(t)θ0 denote the solution.
Theorem(C, Tarfulea, Vicol, ’13). ∃!X ⊂ H1,
limt→∞
distH1 (S(t)θ0,X ) = 0,
uniform for θ0 in bounded sets, X invariant S(t)X = X, compact, asnice (C∞) as forces permit, and dF (X ) <∞.
![Page 45: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/45.jpg)
Forced Critical SQG
SQG+f
∂tθ + (R⊥θ) · ∇θ + Λθ = f
in T2, with θ0 ∈ H1,
H1(T2) = θ |
T2θdx = 0,
T2|∇θ|2dx <∞
f ∈ H1 ∩ L∞. Let S(t)θ0 denote the solution.
Theorem(C, Tarfulea, Vicol, ’13). ∃!X ⊂ H1,
limt→∞
distH1 (S(t)θ0,X ) = 0,
uniform for θ0 in bounded sets, X invariant S(t)X = X, compact, asnice (C∞) as forces permit, and dF (X ) <∞.
![Page 46: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/46.jpg)
Forced Critical SQG
SQG+f
∂tθ + (R⊥θ) · ∇θ + Λθ = f
in T2, with θ0 ∈ H1,
H1(T2) = θ |
T2θdx = 0,
T2|∇θ|2dx <∞
f ∈ H1 ∩ L∞. Let S(t)θ0 denote the solution.
Theorem(C, Tarfulea, Vicol, ’13). ∃!X ⊂ H1,
limt→∞
distH1 (S(t)θ0,X ) = 0,
uniform for θ0 in bounded sets, X invariant S(t)X = X, compact, asnice (C∞) as forces permit, and dF (X ) <∞.
![Page 47: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/47.jpg)
Forced Critical SQG
SQG+f
∂tθ + (R⊥θ) · ∇θ + Λθ = f
in T2, with θ0 ∈ H1,
H1(T2) = θ |
T2θdx = 0,
T2|∇θ|2dx <∞
f ∈ H1 ∩ L∞. Let S(t)θ0 denote the solution.
Theorem(C, Tarfulea, Vicol, ’13). ∃!X ⊂ H1,
limt→∞
distH1 (S(t)θ0,X ) = 0,
uniform for θ0 in bounded sets,
X invariant S(t)X = X, compact, asnice (C∞) as forces permit, and dF (X ) <∞.
![Page 48: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/48.jpg)
Forced Critical SQG
SQG+f
∂tθ + (R⊥θ) · ∇θ + Λθ = f
in T2, with θ0 ∈ H1,
H1(T2) = θ |
T2θdx = 0,
T2|∇θ|2dx <∞
f ∈ H1 ∩ L∞. Let S(t)θ0 denote the solution.
Theorem(C, Tarfulea, Vicol, ’13). ∃!X ⊂ H1,
limt→∞
distH1 (S(t)θ0,X ) = 0,
uniform for θ0 in bounded sets, X invariant S(t)X = X,
compact, asnice (C∞) as forces permit, and dF (X ) <∞.
![Page 49: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/49.jpg)
Forced Critical SQG
SQG+f
∂tθ + (R⊥θ) · ∇θ + Λθ = f
in T2, with θ0 ∈ H1,
H1(T2) = θ |
T2θdx = 0,
T2|∇θ|2dx <∞
f ∈ H1 ∩ L∞. Let S(t)θ0 denote the solution.
Theorem(C, Tarfulea, Vicol, ’13). ∃!X ⊂ H1,
limt→∞
distH1 (S(t)θ0,X ) = 0,
uniform for θ0 in bounded sets, X invariant S(t)X = X, compact, asnice (C∞) as forces permit, and
dF (X ) <∞.
![Page 50: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/50.jpg)
Forced Critical SQG
SQG+f
∂tθ + (R⊥θ) · ∇θ + Λθ = f
in T2, with θ0 ∈ H1,
H1(T2) = θ |
T2θdx = 0,
T2|∇θ|2dx <∞
f ∈ H1 ∩ L∞. Let S(t)θ0 denote the solution.
Theorem(C, Tarfulea, Vicol, ’13). ∃!X ⊂ H1,
limt→∞
distH1 (S(t)θ0,X ) = 0,
uniform for θ0 in bounded sets, X invariant S(t)X = X, compact, asnice (C∞) as forces permit, and dF (X ) <∞.
![Page 51: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/51.jpg)
Periodic case
I Poisson summation gives kernel for Riesz transform
I Poisson summation gives kernel for Λ
I Nonlinear max principle depends on L∞ aloneI Nonlinear fractional Poincare gives absorbing ball in L∞
Lemma(C-Glatt-Holtz-Vicol) There exists a constant C such that for everyp ≥ 2 even, and every φ,
T2 φ = 0 it holds that
T2φp−1Λφdx ≥ 1
p‖Λ 1
2 (φp2 )‖2
L2 + C‖θ‖pLp
Forgetting initial data: long time L∞ bound depends on forces only.Nonlinear max principle gives (small α) absorbing ball in Cα with sizedepending on forces only. Nonlinear max principle based on Cα
gives H1 absorbing ball (higher regularity as well)
![Page 52: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/52.jpg)
Periodic case
I Poisson summation gives kernel for Riesz transformI Poisson summation gives kernel for Λ
I Nonlinear max principle depends on L∞ aloneI Nonlinear fractional Poincare gives absorbing ball in L∞
Lemma(C-Glatt-Holtz-Vicol) There exists a constant C such that for everyp ≥ 2 even, and every φ,
T2 φ = 0 it holds that
T2φp−1Λφdx ≥ 1
p‖Λ 1
2 (φp2 )‖2
L2 + C‖θ‖pLp
Forgetting initial data: long time L∞ bound depends on forces only.Nonlinear max principle gives (small α) absorbing ball in Cα with sizedepending on forces only. Nonlinear max principle based on Cα
gives H1 absorbing ball (higher regularity as well)
![Page 53: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/53.jpg)
Periodic case
I Poisson summation gives kernel for Riesz transformI Poisson summation gives kernel for Λ
I Nonlinear max principle depends on L∞ alone
I Nonlinear fractional Poincare gives absorbing ball in L∞
Lemma(C-Glatt-Holtz-Vicol) There exists a constant C such that for everyp ≥ 2 even, and every φ,
T2 φ = 0 it holds that
T2φp−1Λφdx ≥ 1
p‖Λ 1
2 (φp2 )‖2
L2 + C‖θ‖pLp
Forgetting initial data: long time L∞ bound depends on forces only.Nonlinear max principle gives (small α) absorbing ball in Cα with sizedepending on forces only. Nonlinear max principle based on Cα
gives H1 absorbing ball (higher regularity as well)
![Page 54: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/54.jpg)
Periodic case
I Poisson summation gives kernel for Riesz transformI Poisson summation gives kernel for Λ
I Nonlinear max principle depends on L∞ aloneI Nonlinear fractional Poincare gives absorbing ball in L∞
Lemma(C-Glatt-Holtz-Vicol) There exists a constant C such that for everyp ≥ 2 even, and every φ,
T2 φ = 0 it holds that
T2φp−1Λφdx ≥ 1
p‖Λ 1
2 (φp2 )‖2
L2 + C‖θ‖pLp
Forgetting initial data: long time L∞ bound depends on forces only.Nonlinear max principle gives (small α) absorbing ball in Cα with sizedepending on forces only. Nonlinear max principle based on Cα
gives H1 absorbing ball (higher regularity as well)
![Page 55: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/55.jpg)
Periodic case
I Poisson summation gives kernel for Riesz transformI Poisson summation gives kernel for Λ
I Nonlinear max principle depends on L∞ aloneI Nonlinear fractional Poincare gives absorbing ball in L∞
Lemma(C-Glatt-Holtz-Vicol) There exists a constant C such that for everyp ≥ 2 even, and every φ,
T2 φ = 0 it holds that
T2φp−1Λφdx ≥ 1
p‖Λ 1
2 (φp2 )‖2
L2 + C‖θ‖pLp
Forgetting initial data: long time L∞ bound depends on forces only.Nonlinear max principle gives (small α) absorbing ball in Cα with sizedepending on forces only. Nonlinear max principle based on Cα
gives H1 absorbing ball (higher regularity as well)
![Page 56: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/56.jpg)
Periodic case
I Poisson summation gives kernel for Riesz transformI Poisson summation gives kernel for Λ
I Nonlinear max principle depends on L∞ aloneI Nonlinear fractional Poincare gives absorbing ball in L∞
Lemma(C-Glatt-Holtz-Vicol) There exists a constant C such that for everyp ≥ 2 even, and every φ,
T2 φ = 0 it holds that
T2φp−1Λφdx ≥ 1
p‖Λ 1
2 (φp2 )‖2
L2 + C‖θ‖pLp
Forgetting initial data: long time L∞ bound depends on forces only.
Nonlinear max principle gives (small α) absorbing ball in Cα with sizedepending on forces only. Nonlinear max principle based on Cα
gives H1 absorbing ball (higher regularity as well)
![Page 57: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/57.jpg)
Periodic case
I Poisson summation gives kernel for Riesz transformI Poisson summation gives kernel for Λ
I Nonlinear max principle depends on L∞ aloneI Nonlinear fractional Poincare gives absorbing ball in L∞
Lemma(C-Glatt-Holtz-Vicol) There exists a constant C such that for everyp ≥ 2 even, and every φ,
T2 φ = 0 it holds that
T2φp−1Λφdx ≥ 1
p‖Λ 1
2 (φp2 )‖2
L2 + C‖θ‖pLp
Forgetting initial data: long time L∞ bound depends on forces only.Nonlinear max principle gives (small α) absorbing ball in Cα with sizedepending on forces only.
Nonlinear max principle based on Cα
gives H1 absorbing ball (higher regularity as well)
![Page 58: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/58.jpg)
Periodic case
I Poisson summation gives kernel for Riesz transformI Poisson summation gives kernel for Λ
I Nonlinear max principle depends on L∞ aloneI Nonlinear fractional Poincare gives absorbing ball in L∞
Lemma(C-Glatt-Holtz-Vicol) There exists a constant C such that for everyp ≥ 2 even, and every φ,
T2 φ = 0 it holds that
T2φp−1Λφdx ≥ 1
p‖Λ 1
2 (φp2 )‖2
L2 + C‖θ‖pLp
Forgetting initial data: long time L∞ bound depends on forces only.Nonlinear max principle gives (small α) absorbing ball in Cα with sizedepending on forces only. Nonlinear max principle based on Cα
gives H1 absorbing ball (higher regularity as well)
![Page 59: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/59.jpg)
In this talk: bounded domains
I Lower bounds for Dirichlet Λ in bounded domains (nonlinear maxprinciple) (Ignatova, C)
I Global solutions for a model of electroconvection (Elgindi,Ignatova, Vicol, C)
I Global Holder bounds for critical SQG in bounded domains(Ignatova, C)
![Page 60: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/60.jpg)
In this talk: bounded domains
I Lower bounds for Dirichlet Λ in bounded domains (nonlinear maxprinciple) (Ignatova, C)
I Global solutions for a model of electroconvection (Elgindi,Ignatova, Vicol, C)
I Global Holder bounds for critical SQG in bounded domains(Ignatova, C)
![Page 61: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/61.jpg)
In this talk: bounded domains
I Lower bounds for Dirichlet Λ in bounded domains (nonlinear maxprinciple) (Ignatova, C)
I Global solutions for a model of electroconvection (Elgindi,Ignatova, Vicol, C)
I Global Holder bounds for critical SQG in bounded domains(Ignatova, C)
![Page 62: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/62.jpg)
In this talk: bounded domains
I Lower bounds for Dirichlet Λ in bounded domains (nonlinear maxprinciple) (Ignatova, C)
I Global solutions for a model of electroconvection (Elgindi,Ignatova, Vicol, C)
I Global Holder bounds for critical SQG in bounded domains(Ignatova, C)
![Page 63: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/63.jpg)
Basics in bounded domainsΩ ⊂ Rd , smooth boundary.
Normalized igenfunctions,
Ω
w2j dx = 1
with homogeneous Dirichlet BC:
−∆wj = λjwj ,
Well-known:0 < λ1 ≤ ... ≤ λj →∞
−∆ is a positive selfadjoint operator in L2(Ω) with domainD (−∆) = H2(Ω) ∩ H1
0 (Ω). The ground state w1 is positive and
c0d(x) ≤ w1(x) ≤ C0d(x),
whered(x) = dist(x , ∂Ω)
![Page 64: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/64.jpg)
Basics in bounded domainsΩ ⊂ Rd , smooth boundary. Normalized igenfunctions,
Ω
w2j dx = 1
with homogeneous Dirichlet BC:
−∆wj = λjwj ,
Well-known:0 < λ1 ≤ ... ≤ λj →∞
−∆ is a positive selfadjoint operator in L2(Ω) with domainD (−∆) = H2(Ω) ∩ H1
0 (Ω). The ground state w1 is positive and
c0d(x) ≤ w1(x) ≤ C0d(x),
whered(x) = dist(x , ∂Ω)
![Page 65: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/65.jpg)
Basics in bounded domainsΩ ⊂ Rd , smooth boundary. Normalized igenfunctions,
Ω
w2j dx = 1
with homogeneous Dirichlet BC:
−∆wj = λjwj ,
Well-known:0 < λ1 ≤ ... ≤ λj →∞
−∆ is a positive selfadjoint operator in L2(Ω) with domainD (−∆) = H2(Ω) ∩ H1
0 (Ω). The ground state w1 is positive and
c0d(x) ≤ w1(x) ≤ C0d(x),
whered(x) = dist(x , ∂Ω)
![Page 66: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/66.jpg)
Basics in bounded domainsΩ ⊂ Rd , smooth boundary. Normalized igenfunctions,
Ω
w2j dx = 1
with homogeneous Dirichlet BC:
−∆wj = λjwj ,
Well-known:0 < λ1 ≤ ... ≤ λj →∞
−∆ is a positive selfadjoint operator in L2(Ω) with domainD (−∆) = H2(Ω) ∩ H1
0 (Ω).
The ground state w1 is positive and
c0d(x) ≤ w1(x) ≤ C0d(x),
whered(x) = dist(x , ∂Ω)
![Page 67: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/67.jpg)
Basics in bounded domainsΩ ⊂ Rd , smooth boundary. Normalized igenfunctions,
Ω
w2j dx = 1
with homogeneous Dirichlet BC:
−∆wj = λjwj ,
Well-known:0 < λ1 ≤ ... ≤ λj →∞
−∆ is a positive selfadjoint operator in L2(Ω) with domainD (−∆) = H2(Ω) ∩ H1
0 (Ω). The ground state w1 is positive and
c0d(x) ≤ w1(x) ≤ C0d(x),
whered(x) = dist(x , ∂Ω)
![Page 68: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/68.jpg)
Functional Calculus
(−∆)α f =∞∑j=1
λαj fjwj
withfj =
Ω
f (y)wj (y)dy
and for f ∈ D ((−∆)α) = f | (λαj fj ) ∈ `2(N). We denote by
ΛD = (−∆)12
It is well-known and easy to show that (Kato conjecture, trivial case)
D (ΛD) = H10 (Ω).
Indeed, for f ∈ D (−∆) we have
‖∇f‖2L2(Ω) =
Ω
f (−∆) fdx = ‖ΛDf‖2L2(Ω).
![Page 69: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/69.jpg)
Fractional powers in terms of heat kernel
No explicit formulas for kernels.
Identity
λα = cα ∞
0(1− e−tλ)t−1−αdt ,
with1 = cα
∞0
(1− e−s)s−1−αds,
valid for 0 ≤ α < 1. Representation
(Λ2αD f )(x) = ((−∆)α f ) (x) = cα
∞0
[f (x)− et∆f (x)
]t−1−αdt
for f ∈ D ((−∆)α) = D(
(−ΛD)2α)
.
![Page 70: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/70.jpg)
Fractional powers in terms of heat kernel
No explicit formulas for kernels. Identity
λα = cα ∞
0(1− e−tλ)t−1−αdt ,
with1 = cα
∞0
(1− e−s)s−1−αds,
valid for 0 ≤ α < 1.
Representation
(Λ2αD f )(x) = ((−∆)α f ) (x) = cα
∞0
[f (x)− et∆f (x)
]t−1−αdt
for f ∈ D ((−∆)α) = D(
(−ΛD)2α)
.
![Page 71: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/71.jpg)
Fractional powers in terms of heat kernel
No explicit formulas for kernels. Identity
λα = cα ∞
0(1− e−tλ)t−1−αdt ,
with1 = cα
∞0
(1− e−s)s−1−αds,
valid for 0 ≤ α < 1. Representation
(Λ2αD f )(x) = ((−∆)α f ) (x) = cα
∞0
[f (x)− et∆f (x)
]t−1−αdt
for f ∈ D ((−∆)α) = D(
(−ΛD)2α)
.
![Page 72: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/72.jpg)
Electroconvection
Electric field determined by charge density:∇× E = 0,∇ · E = ρ,
in Q ⊂ R3. Boundary conditions at ∂Q. Charge density ρ confined todomain Ω ⊂ Q:
ρ = 2qδΩ
carried by a flow in Ω with ε anisotropic permittivity (an operator).Conducting fluid confined to domain Ω:
∂tq +∇ · (uq + εE) = 0,∂tu + u · ∇u − ν∆u +∇p = qεE , ∇ · u = 0.
Ω ⊂ R2 × 0 (thin film): Fractional Laplacian emerges.
![Page 73: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/73.jpg)
Electroconvection
Electric field determined by charge density:∇× E = 0,∇ · E = ρ,
in Q ⊂ R3. Boundary conditions at ∂Q. Charge density ρ confined todomain Ω ⊂ Q:
ρ = 2qδΩ
carried by a flow in Ω with ε anisotropic permittivity (an operator).Conducting fluid confined to domain Ω:
∂tq +∇ · (uq + εE) = 0,∂tu + u · ∇u − ν∆u +∇p = qεE , ∇ · u = 0.
Ω ⊂ R2 × 0 (thin film): Fractional Laplacian emerges.
![Page 74: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/74.jpg)
Electroconvection
Electric field determined by charge density:∇× E = 0,∇ · E = ρ,
in Q ⊂ R3. Boundary conditions at ∂Q. Charge density ρ confined todomain Ω ⊂ Q:
ρ = 2qδΩ
carried by a flow in Ω with ε anisotropic permittivity (an operator).
Conducting fluid confined to domain Ω:∂tq +∇ · (uq + εE) = 0,∂tu + u · ∇u − ν∆u +∇p = qεE , ∇ · u = 0.
Ω ⊂ R2 × 0 (thin film): Fractional Laplacian emerges.
![Page 75: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/75.jpg)
Electroconvection
Electric field determined by charge density:∇× E = 0,∇ · E = ρ,
in Q ⊂ R3. Boundary conditions at ∂Q. Charge density ρ confined todomain Ω ⊂ Q:
ρ = 2qδΩ
carried by a flow in Ω with ε anisotropic permittivity (an operator).Conducting fluid confined to domain Ω:
∂tq +∇ · (uq + εE) = 0,∂tu + u · ∇u − ν∆u +∇p = qεE , ∇ · u = 0.
Ω ⊂ R2 × 0 (thin film): Fractional Laplacian emerges.
![Page 76: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/76.jpg)
Electroconvection
Electric field determined by charge density:∇× E = 0,∇ · E = ρ,
in Q ⊂ R3. Boundary conditions at ∂Q. Charge density ρ confined todomain Ω ⊂ Q:
ρ = 2qδΩ
carried by a flow in Ω with ε anisotropic permittivity (an operator).Conducting fluid confined to domain Ω:
∂tq +∇ · (uq + εE) = 0,∂tu + u · ∇u − ν∆u +∇p = qεE , ∇ · u = 0.
Ω ⊂ R2 × 0 (thin film): Fractional Laplacian emerges.
![Page 77: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/77.jpg)
Electroconvection example
Thin film of fluid = 2DNS in fluid region Ω ⊂ R2 × 0.
Electrodsshare boundaries with ∂Ω. Two connected components of ∂Ω kept attwo different voltages, V and 0. Electric field
E = −∇Φ
defined in Q = Ω×R with inhomogeneous boundary conditions for Φ.
−∆3Φ = 2qδΩ, Φ∂Q = V , 0.
Solved by
Φ(x , z) = Φ0(x) +
e−zΛD Λ−1
D q, z > 0,ezΛD Λ−1
D q, z < 0
PermittivityεE = (−∂1Φ,−∂2Φ,0)|Ω
![Page 78: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/78.jpg)
Electroconvection example
Thin film of fluid = 2DNS in fluid region Ω ⊂ R2 × 0. Electrodsshare boundaries with ∂Ω.
Two connected components of ∂Ω kept attwo different voltages, V and 0. Electric field
E = −∇Φ
defined in Q = Ω×R with inhomogeneous boundary conditions for Φ.
−∆3Φ = 2qδΩ, Φ∂Q = V , 0.
Solved by
Φ(x , z) = Φ0(x) +
e−zΛD Λ−1
D q, z > 0,ezΛD Λ−1
D q, z < 0
PermittivityεE = (−∂1Φ,−∂2Φ,0)|Ω
![Page 79: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/79.jpg)
Electroconvection example
Thin film of fluid = 2DNS in fluid region Ω ⊂ R2 × 0. Electrodsshare boundaries with ∂Ω. Two connected components of ∂Ω kept attwo different voltages, V and 0. Electric field
E = −∇Φ
defined in Q = Ω×R with inhomogeneous boundary conditions for Φ.
−∆3Φ = 2qδΩ, Φ∂Q = V , 0.
Solved by
Φ(x , z) = Φ0(x) +
e−zΛD Λ−1
D q, z > 0,ezΛD Λ−1
D q, z < 0
PermittivityεE = (−∂1Φ,−∂2Φ,0)|Ω
![Page 80: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/80.jpg)
Electroconvection example
Thin film of fluid = 2DNS in fluid region Ω ⊂ R2 × 0. Electrodsshare boundaries with ∂Ω. Two connected components of ∂Ω kept attwo different voltages, V and 0. Electric field
E = −∇Φ
defined in Q = Ω×R with inhomogeneous boundary conditions for Φ.
−∆3Φ = 2qδΩ, Φ∂Q = V , 0.
Solved by
Φ(x , z) = Φ0(x) +
e−zΛD Λ−1
D q, z > 0,ezΛD Λ−1
D q, z < 0
PermittivityεE = (−∂1Φ,−∂2Φ,0)|Ω
![Page 81: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/81.jpg)
Electroconvection example
Thin film of fluid = 2DNS in fluid region Ω ⊂ R2 × 0. Electrodsshare boundaries with ∂Ω. Two connected components of ∂Ω kept attwo different voltages, V and 0. Electric field
E = −∇Φ
defined in Q = Ω×R with inhomogeneous boundary conditions for Φ.
−∆3Φ = 2qδΩ, Φ∂Q = V , 0.
Solved by
Φ(x , z) = Φ0(x) +
e−zΛD Λ−1
D q, z > 0,ezΛD Λ−1
D q, z < 0
PermittivityεE = (−∂1Φ,−∂2Φ,0)|Ω
![Page 82: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/82.jpg)
Global Regularity for Electroconvection in 2D BoundedDomains
Theorem(C, Elgindi, Ignatova, Vicol) Let Ω ⊂ R2 open, bounded, with smoothboundary. Let u0 ∈ [H1
0 (Ω) ∩ H2(Ω)]2 be divergence-free. Letq0 ∈ H1
0 (Ω) ∩ H2(Ω). Then the electroconvection system ∂tu + u · ∇u +∇p = ν∆u − q∇Λ−1D q − q∇Φ0,
divu = 0,∂tq + u · ∇q + ΛDq = 0
with homogeneous Dirichlet boundary conditions for both u and qhas global unique strong solutions,
u ∈ L∞(0,T ; [H10 (Ω) ∩ H2(Ω)]2) ∩ L2(0,T ; H
52 (Ω)2),
q ∈ L∞(0,T ; W 1,40 (Ω) ∩ H2(Ω)) ∩ L2(0,T ; H
52 (Ω)),
![Page 83: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/83.jpg)
Critical SQG in bounded domains
∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0
with RD = ∇Λ−1D . Ω ⊂ Rd bounded domain with smooth boundary.
Global weak solutions:
Theorem(C, Ignatova) Let θ0 ∈ L2(Ω) and let T > 0. There exists a weaksolution of critical SQG,
θ ∈ L∞(0,T ; L2(Ω)) ∩ L2(
(0,T ;D(
Λ12D
))satisfying limt→0θ(t) = θ0 weakly in L2(Ω).Local existence of smooth solutions: OK as well.
![Page 84: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/84.jpg)
Critical SQG in bounded domains
∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0
with RD = ∇Λ−1D .
Ω ⊂ Rd bounded domain with smooth boundary.Global weak solutions:
Theorem(C, Ignatova) Let θ0 ∈ L2(Ω) and let T > 0. There exists a weaksolution of critical SQG,
θ ∈ L∞(0,T ; L2(Ω)) ∩ L2(
(0,T ;D(
Λ12D
))satisfying limt→0θ(t) = θ0 weakly in L2(Ω).Local existence of smooth solutions: OK as well.
![Page 85: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/85.jpg)
Critical SQG in bounded domains
∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0
with RD = ∇Λ−1D . Ω ⊂ Rd bounded domain with smooth boundary.
Global weak solutions:
Theorem(C, Ignatova) Let θ0 ∈ L2(Ω) and let T > 0. There exists a weaksolution of critical SQG,
θ ∈ L∞(0,T ; L2(Ω)) ∩ L2(
(0,T ;D(
Λ12D
))satisfying limt→0θ(t) = θ0 weakly in L2(Ω).Local existence of smooth solutions: OK as well.
![Page 86: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/86.jpg)
Critical SQG in bounded domains
∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0
with RD = ∇Λ−1D . Ω ⊂ Rd bounded domain with smooth boundary.
Global weak solutions:
Theorem(C, Ignatova) Let θ0 ∈ L2(Ω) and let T > 0. There exists a weaksolution of critical SQG,
θ ∈ L∞(0,T ; L2(Ω)) ∩ L2(
(0,T ;D(
Λ12D
))satisfying limt→0θ(t) = θ0 weakly in L2(Ω).
Local existence of smooth solutions: OK as well.
![Page 87: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/87.jpg)
Critical SQG in bounded domains
∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0
with RD = ∇Λ−1D . Ω ⊂ Rd bounded domain with smooth boundary.
Global weak solutions:
Theorem(C, Ignatova) Let θ0 ∈ L2(Ω) and let T > 0. There exists a weaksolution of critical SQG,
θ ∈ L∞(0,T ; L2(Ω)) ∩ L2(
(0,T ;D(
Λ12D
))satisfying limt→0θ(t) = θ0 weakly in L2(Ω).Local existence of smooth solutions: OK as well.
![Page 88: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/88.jpg)
Holder bounds for critical SQG in bounded domains
Theorem(C, Ignatova) Let θ(x , t) be a smooth solution of
∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0
in the smooth bounded domain Ω. There exists 0 < α < 1 dependingonly on ‖θ0‖L∞(Ω), and a constant Γ > 0 depending on the domain Ωsuch that, for any ` > 0 sufficiently small
supd(x)≥`, |h|≤ `
16 , t≥0
|θ(x + h, t)− θ(x , t)||h|α
≤ ‖θ0‖Cα + Γ`−α‖θ0‖L∞(Ω)
holds.recall: d(x) = dist(x , ∂Ω). Proof based on adequate nonlinear maxbounds and commutators.Higher interior regularity and global existence follow.
![Page 89: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/89.jpg)
Holder bounds for critical SQG in bounded domains
Theorem(C, Ignatova) Let θ(x , t) be a smooth solution of
∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0
in the smooth bounded domain Ω. There exists 0 < α < 1 dependingonly on ‖θ0‖L∞(Ω), and a constant Γ > 0 depending on the domain Ωsuch that, for any ` > 0 sufficiently small
supd(x)≥`, |h|≤ `
16 , t≥0
|θ(x + h, t)− θ(x , t)||h|α
≤ ‖θ0‖Cα + Γ`−α‖θ0‖L∞(Ω)
holds.
recall: d(x) = dist(x , ∂Ω). Proof based on adequate nonlinear maxbounds and commutators.Higher interior regularity and global existence follow.
![Page 90: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/90.jpg)
Holder bounds for critical SQG in bounded domains
Theorem(C, Ignatova) Let θ(x , t) be a smooth solution of
∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0
in the smooth bounded domain Ω. There exists 0 < α < 1 dependingonly on ‖θ0‖L∞(Ω), and a constant Γ > 0 depending on the domain Ωsuch that, for any ` > 0 sufficiently small
supd(x)≥`, |h|≤ `
16 , t≥0
|θ(x + h, t)− θ(x , t)||h|α
≤ ‖θ0‖Cα + Γ`−α‖θ0‖L∞(Ω)
holds.recall: d(x) = dist(x , ∂Ω).
Proof based on adequate nonlinear maxbounds and commutators.Higher interior regularity and global existence follow.
![Page 91: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/91.jpg)
Holder bounds for critical SQG in bounded domains
Theorem(C, Ignatova) Let θ(x , t) be a smooth solution of
∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0
in the smooth bounded domain Ω. There exists 0 < α < 1 dependingonly on ‖θ0‖L∞(Ω), and a constant Γ > 0 depending on the domain Ωsuch that, for any ` > 0 sufficiently small
supd(x)≥`, |h|≤ `
16 , t≥0
|θ(x + h, t)− θ(x , t)||h|α
≤ ‖θ0‖Cα + Γ`−α‖θ0‖L∞(Ω)
holds.recall: d(x) = dist(x , ∂Ω). Proof based on adequate nonlinear maxbounds and commutators.
Higher interior regularity and global existence follow.
![Page 92: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/92.jpg)
Holder bounds for critical SQG in bounded domains
Theorem(C, Ignatova) Let θ(x , t) be a smooth solution of
∂tθ + (R⊥D θ) · ∇θ + ΛDθ = 0
in the smooth bounded domain Ω. There exists 0 < α < 1 dependingonly on ‖θ0‖L∞(Ω), and a constant Γ > 0 depending on the domain Ωsuch that, for any ` > 0 sufficiently small
supd(x)≥`, |h|≤ `
16 , t≥0
|θ(x + h, t)− θ(x , t)||h|α
≤ ‖θ0‖Cα + Γ`−α‖θ0‖L∞(Ω)
holds.recall: d(x) = dist(x , ∂Ω). Proof based on adequate nonlinear maxbounds and commutators.Higher interior regularity and global existence follow.
![Page 93: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/93.jpg)
Elements of the proof
I Gaussian bounds for heat kernel; cancellation due to translationinvariance effective for small time.
I Nonlinear maximum principle (lower bound for ΛD) givingsmoothing and a strong boundary repulsion damping effect.
I Good cutoff χ and bound for the commutator [δh,ΛD] away fromboundary; (the most expensive item, fighting boundary repulsion)
I Finite difference bounds for Riesz transforms using the nonlinearmax principle bound in its finite difference variant.
![Page 94: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/94.jpg)
Elements of the proof
I Gaussian bounds for heat kernel; cancellation due to translationinvariance effective for small time.
I Nonlinear maximum principle (lower bound for ΛD) givingsmoothing and a strong boundary repulsion damping effect.
I Good cutoff χ and bound for the commutator [δh,ΛD] away fromboundary; (the most expensive item, fighting boundary repulsion)
I Finite difference bounds for Riesz transforms using the nonlinearmax principle bound in its finite difference variant.
![Page 95: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/95.jpg)
Elements of the proof
I Gaussian bounds for heat kernel; cancellation due to translationinvariance effective for small time.
I Nonlinear maximum principle (lower bound for ΛD) givingsmoothing and a strong boundary repulsion damping effect.
I Good cutoff χ and bound for the commutator [δh,ΛD] away fromboundary; (the most expensive item, fighting boundary repulsion)
I Finite difference bounds for Riesz transforms using the nonlinearmax principle bound in its finite difference variant.
![Page 96: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/96.jpg)
Elements of the proof
I Gaussian bounds for heat kernel; cancellation due to translationinvariance effective for small time.
I Nonlinear maximum principle (lower bound for ΛD) givingsmoothing and a strong boundary repulsion damping effect.
I Good cutoff χ and bound for the commutator [δh,ΛD] away fromboundary; (the most expensive item, fighting boundary repulsion)
I Finite difference bounds for Riesz transforms using the nonlinearmax principle bound in its finite difference variant.
![Page 97: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/97.jpg)
Bounds for heat kernelWe use precise upper and lower bounds for the kernel HD(t , x , y) ofthe heat operator,
(et∆f )(x) =
Ω
HD(t , x , y)f (y)dy .
These are as follows (Davies, Q.S Zhang): There exists a timeT > 0 depending on the domain Ω and constants c, C, k , K ,depending on T and Ω such that
c min(
w1(x)|x−y| ,1
)min
(w1(y)|x−y| ,1
)t−
d2 e−
|x−y|2kt ≤
HD(t , x , y) ≤ C min(
w1(x)|x−y| ,1
)min
(w1(y)|x−y| ,1
)t−
d2 e−
|x−y|2Kt
holds for all 0 ≤ t ≤ T . Moreover
|∇xHD(t , x , y)|HD(t , x , y)
≤ C
1d(x) , if
√t ≥ d(x),
1√t
(1 + |x−y|√
t
), if√
t ≤ d(x)
holds for all 0 ≤ t ≤ T .
![Page 98: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/98.jpg)
Bounds for heat kernelWe use precise upper and lower bounds for the kernel HD(t , x , y) ofthe heat operator,
(et∆f )(x) =
Ω
HD(t , x , y)f (y)dy .
These are as follows (Davies, Q.S Zhang):
There exists a timeT > 0 depending on the domain Ω and constants c, C, k , K ,depending on T and Ω such that
c min(
w1(x)|x−y| ,1
)min
(w1(y)|x−y| ,1
)t−
d2 e−
|x−y|2kt ≤
HD(t , x , y) ≤ C min(
w1(x)|x−y| ,1
)min
(w1(y)|x−y| ,1
)t−
d2 e−
|x−y|2Kt
holds for all 0 ≤ t ≤ T . Moreover
|∇xHD(t , x , y)|HD(t , x , y)
≤ C
1d(x) , if
√t ≥ d(x),
1√t
(1 + |x−y|√
t
), if√
t ≤ d(x)
holds for all 0 ≤ t ≤ T .
![Page 99: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/99.jpg)
Bounds for heat kernelWe use precise upper and lower bounds for the kernel HD(t , x , y) ofthe heat operator,
(et∆f )(x) =
Ω
HD(t , x , y)f (y)dy .
These are as follows (Davies, Q.S Zhang): There exists a timeT > 0 depending on the domain Ω and constants c, C, k , K ,depending on T and Ω such that
c min(
w1(x)|x−y| ,1
)min
(w1(y)|x−y| ,1
)t−
d2 e−
|x−y|2kt ≤
HD(t , x , y) ≤ C min(
w1(x)|x−y| ,1
)min
(w1(y)|x−y| ,1
)t−
d2 e−
|x−y|2Kt
holds for all 0 ≤ t ≤ T .
Moreover
|∇xHD(t , x , y)|HD(t , x , y)
≤ C
1d(x) , if
√t ≥ d(x),
1√t
(1 + |x−y|√
t
), if√
t ≤ d(x)
holds for all 0 ≤ t ≤ T .
![Page 100: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/100.jpg)
Bounds for heat kernelWe use precise upper and lower bounds for the kernel HD(t , x , y) ofthe heat operator,
(et∆f )(x) =
Ω
HD(t , x , y)f (y)dy .
These are as follows (Davies, Q.S Zhang): There exists a timeT > 0 depending on the domain Ω and constants c, C, k , K ,depending on T and Ω such that
c min(
w1(x)|x−y| ,1
)min
(w1(y)|x−y| ,1
)t−
d2 e−
|x−y|2kt ≤
HD(t , x , y) ≤ C min(
w1(x)|x−y| ,1
)min
(w1(y)|x−y| ,1
)t−
d2 e−
|x−y|2Kt
holds for all 0 ≤ t ≤ T . Moreover
|∇xHD(t , x , y)|HD(t , x , y)
≤ C
1d(x) , if
√t ≥ d(x),
1√t
(1 + |x−y|√
t
), if√
t ≤ d(x)
holds for all 0 ≤ t ≤ T .
![Page 101: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/101.jpg)
Symmetry of bounds for the gradientNote that, in view of
HD(t , x , y) =∞∑j=1
e−tλj wj (x)wj (y)
elliptic regularity estimates and Sobolev embedding which implyuniform absolute convergence of the series (if ∂Ω is smooth enough),we have that
∂β1 HD(t , y , x) = ∂β2 HD(t , x , y) =∞∑j=1
e−tλj∂βx wj (y)wj (x)
for positive t , where we denoted by ∂β1 and ∂β2 derivatives with respectto the first spatial variables, respectively the second spatial variables.
Therefore, the previous gradient bounds result in
|∇y HD(t , x , y)|HD(t , x , y)
≤ C
1d(y) , if
√t ≥ d(y),
1√t
(1 + |x−y|√
t
), if√
t ≤ d(y)
![Page 102: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/102.jpg)
Symmetry of bounds for the gradientNote that, in view of
HD(t , x , y) =∞∑j=1
e−tλj wj (x)wj (y)
elliptic regularity estimates and Sobolev embedding which implyuniform absolute convergence of the series (if ∂Ω is smooth enough),we have that
∂β1 HD(t , y , x) = ∂β2 HD(t , x , y) =∞∑j=1
e−tλj∂βx wj (y)wj (x)
for positive t , where we denoted by ∂β1 and ∂β2 derivatives with respectto the first spatial variables, respectively the second spatial variables.Therefore, the previous gradient bounds result in
|∇y HD(t , x , y)|HD(t , x , y)
≤ C
1d(y) , if
√t ≥ d(y),
1√t
(1 + |x−y|√
t
), if√
t ≤ d(y)
![Page 103: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/103.jpg)
Additional bounds; translation invariance effect
|∇x∇xHD(x , y , t)| ≤ Ct−1− d2 e−
|x−y|2
Kt
holds for t ≤ cd(x)2 and 0 < t ≤ T .
This follows from previousbounds. Important additional bounds we need are
I1(x , t) =
Ω
|(∇x +∇y )HD(x , y , t)|dy ≤ Ct−12 e−
d(x)2
Kt
and
I2(x , t) =
Ω
|∇x (∇x +∇y )HD(x , y , t)|dy ≤ Ct−1e−d(x)2
Kt
valid for t ≤ cd(x)2. These bounds reflect the fact that translationinvariance is remembered in the solution of the heat equation withDirichlet boundary data for short time, away from the boundary. Theyimply that T
0t−
k2 Ij (x , t)dt ≤ d(x)2−j−k
for j = 1,2 and k ≥ 0.
![Page 104: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/104.jpg)
Additional bounds; translation invariance effect
|∇x∇xHD(x , y , t)| ≤ Ct−1− d2 e−
|x−y|2
Kt
holds for t ≤ cd(x)2 and 0 < t ≤ T . This follows from previousbounds.
Important additional bounds we need are
I1(x , t) =
Ω
|(∇x +∇y )HD(x , y , t)|dy ≤ Ct−12 e−
d(x)2
Kt
and
I2(x , t) =
Ω
|∇x (∇x +∇y )HD(x , y , t)|dy ≤ Ct−1e−d(x)2
Kt
valid for t ≤ cd(x)2. These bounds reflect the fact that translationinvariance is remembered in the solution of the heat equation withDirichlet boundary data for short time, away from the boundary. Theyimply that T
0t−
k2 Ij (x , t)dt ≤ d(x)2−j−k
for j = 1,2 and k ≥ 0.
![Page 105: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/105.jpg)
Additional bounds; translation invariance effect
|∇x∇xHD(x , y , t)| ≤ Ct−1− d2 e−
|x−y|2
Kt
holds for t ≤ cd(x)2 and 0 < t ≤ T . This follows from previousbounds. Important additional bounds we need are
I1(x , t) =
Ω
|(∇x +∇y )HD(x , y , t)|dy ≤ Ct−12 e−
d(x)2
Kt
and
I2(x , t) =
Ω
|∇x (∇x +∇y )HD(x , y , t)|dy ≤ Ct−1e−d(x)2
Kt
valid for t ≤ cd(x)2. These bounds reflect the fact that translationinvariance is remembered in the solution of the heat equation withDirichlet boundary data for short time, away from the boundary. Theyimply that T
0t−
k2 Ij (x , t)dt ≤ d(x)2−j−k
for j = 1,2 and k ≥ 0.
![Page 106: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/106.jpg)
Additional bounds; translation invariance effect
|∇x∇xHD(x , y , t)| ≤ Ct−1− d2 e−
|x−y|2
Kt
holds for t ≤ cd(x)2 and 0 < t ≤ T . This follows from previousbounds. Important additional bounds we need are
I1(x , t) =
Ω
|(∇x +∇y )HD(x , y , t)|dy ≤ Ct−12 e−
d(x)2
Kt
and
I2(x , t) =
Ω
|∇x (∇x +∇y )HD(x , y , t)|dy ≤ Ct−1e−d(x)2
Kt
valid for t ≤ cd(x)2.
These bounds reflect the fact that translationinvariance is remembered in the solution of the heat equation withDirichlet boundary data for short time, away from the boundary. Theyimply that T
0t−
k2 Ij (x , t)dt ≤ d(x)2−j−k
for j = 1,2 and k ≥ 0.
![Page 107: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/107.jpg)
Additional bounds; translation invariance effect
|∇x∇xHD(x , y , t)| ≤ Ct−1− d2 e−
|x−y|2
Kt
holds for t ≤ cd(x)2 and 0 < t ≤ T . This follows from previousbounds. Important additional bounds we need are
I1(x , t) =
Ω
|(∇x +∇y )HD(x , y , t)|dy ≤ Ct−12 e−
d(x)2
Kt
and
I2(x , t) =
Ω
|∇x (∇x +∇y )HD(x , y , t)|dy ≤ Ct−1e−d(x)2
Kt
valid for t ≤ cd(x)2. These bounds reflect the fact that translationinvariance is remembered in the solution of the heat equation withDirichlet boundary data for short time, away from the boundary.
Theyimply that T
0t−
k2 Ij (x , t)dt ≤ d(x)2−j−k
for j = 1,2 and k ≥ 0.
![Page 108: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/108.jpg)
Additional bounds; translation invariance effect
|∇x∇xHD(x , y , t)| ≤ Ct−1− d2 e−
|x−y|2
Kt
holds for t ≤ cd(x)2 and 0 < t ≤ T . This follows from previousbounds. Important additional bounds we need are
I1(x , t) =
Ω
|(∇x +∇y )HD(x , y , t)|dy ≤ Ct−12 e−
d(x)2
Kt
and
I2(x , t) =
Ω
|∇x (∇x +∇y )HD(x , y , t)|dy ≤ Ct−1e−d(x)2
Kt
valid for t ≤ cd(x)2. These bounds reflect the fact that translationinvariance is remembered in the solution of the heat equation withDirichlet boundary data for short time, away from the boundary. Theyimply that T
0t−
k2 Ij (x , t)dt ≤ d(x)2−j−k
for j = 1,2 and k ≥ 0.
![Page 109: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/109.jpg)
The convex damping inequality
Proposition(C, Ignatova) Let Ω be a bounded domain with smooth boundary, let0 < s < 2. There exists a constant C depending on the domain andon s such that for every Φ, a C2 convex function satisfying Φ(0) = 0,and every f ∈ C∞0 (Ω)
Φ′(f )ΛsDf − Λs
D(Φ(f )) ≥ Cd(x)s (f (x)Φ′(f (x))− Φ(f (x)))
holds pointwise in Ω.This generalizes the Cordoba-Cordoba inequality from Rd
(d(x) =∞). The proof follows from approximation, convexity andthe fact that Θ = et∆1 obeys 0 ≤ Θ ≤ 1 and
ΛsD1 =
∞0
t−1− s2 (1−Θ(x , t))dt ≥ cd(x)−s
Dramatically different from Rd !
![Page 110: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/110.jpg)
The convex damping inequality
Proposition(C, Ignatova) Let Ω be a bounded domain with smooth boundary, let0 < s < 2. There exists a constant C depending on the domain andon s such that for every Φ, a C2 convex function satisfying Φ(0) = 0,and every f ∈ C∞0 (Ω)
Φ′(f )ΛsDf − Λs
D(Φ(f )) ≥ Cd(x)s (f (x)Φ′(f (x))− Φ(f (x)))
holds pointwise in Ω.
This generalizes the Cordoba-Cordoba inequality from Rd
(d(x) =∞). The proof follows from approximation, convexity andthe fact that Θ = et∆1 obeys 0 ≤ Θ ≤ 1 and
ΛsD1 =
∞0
t−1− s2 (1−Θ(x , t))dt ≥ cd(x)−s
Dramatically different from Rd !
![Page 111: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/111.jpg)
The convex damping inequality
Proposition(C, Ignatova) Let Ω be a bounded domain with smooth boundary, let0 < s < 2. There exists a constant C depending on the domain andon s such that for every Φ, a C2 convex function satisfying Φ(0) = 0,and every f ∈ C∞0 (Ω)
Φ′(f )ΛsDf − Λs
D(Φ(f )) ≥ Cd(x)s (f (x)Φ′(f (x))− Φ(f (x)))
holds pointwise in Ω.This generalizes the Cordoba-Cordoba inequality from Rd
(d(x) =∞).
The proof follows from approximation, convexity andthe fact that Θ = et∆1 obeys 0 ≤ Θ ≤ 1 and
ΛsD1 =
∞0
t−1− s2 (1−Θ(x , t))dt ≥ cd(x)−s
Dramatically different from Rd !
![Page 112: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/112.jpg)
The convex damping inequality
Proposition(C, Ignatova) Let Ω be a bounded domain with smooth boundary, let0 < s < 2. There exists a constant C depending on the domain andon s such that for every Φ, a C2 convex function satisfying Φ(0) = 0,and every f ∈ C∞0 (Ω)
Φ′(f )ΛsDf − Λs
D(Φ(f )) ≥ Cd(x)s (f (x)Φ′(f (x))− Φ(f (x)))
holds pointwise in Ω.This generalizes the Cordoba-Cordoba inequality from Rd
(d(x) =∞). The proof follows from approximation, convexity andthe fact that
Θ = et∆1 obeys 0 ≤ Θ ≤ 1 and
ΛsD1 =
∞0
t−1− s2 (1−Θ(x , t))dt ≥ cd(x)−s
Dramatically different from Rd !
![Page 113: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/113.jpg)
The convex damping inequality
Proposition(C, Ignatova) Let Ω be a bounded domain with smooth boundary, let0 < s < 2. There exists a constant C depending on the domain andon s such that for every Φ, a C2 convex function satisfying Φ(0) = 0,and every f ∈ C∞0 (Ω)
Φ′(f )ΛsDf − Λs
D(Φ(f )) ≥ Cd(x)s (f (x)Φ′(f (x))− Φ(f (x)))
holds pointwise in Ω.This generalizes the Cordoba-Cordoba inequality from Rd
(d(x) =∞). The proof follows from approximation, convexity andthe fact that Θ = et∆1 obeys 0 ≤ Θ ≤ 1 and
ΛsD1 =
∞0
t−1− s2 (1−Θ(x , t))dt ≥ cd(x)−s
Dramatically different from Rd !
![Page 114: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/114.jpg)
The convex damping inequality
Proposition(C, Ignatova) Let Ω be a bounded domain with smooth boundary, let0 < s < 2. There exists a constant C depending on the domain andon s such that for every Φ, a C2 convex function satisfying Φ(0) = 0,and every f ∈ C∞0 (Ω)
Φ′(f )ΛsDf − Λs
D(Φ(f )) ≥ Cd(x)s (f (x)Φ′(f (x))− Φ(f (x)))
holds pointwise in Ω.This generalizes the Cordoba-Cordoba inequality from Rd
(d(x) =∞). The proof follows from approximation, convexity andthe fact that Θ = et∆1 obeys 0 ≤ Θ ≤ 1 and
ΛsD1 =
∞0
t−1− s2 (1−Θ(x , t))dt ≥ cd(x)−s
Dramatically different from Rd !
![Page 115: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/115.jpg)
The convex damping inequality
Proposition(C, Ignatova) Let Ω be a bounded domain with smooth boundary, let0 < s < 2. There exists a constant C depending on the domain andon s such that for every Φ, a C2 convex function satisfying Φ(0) = 0,and every f ∈ C∞0 (Ω)
Φ′(f )ΛsDf − Λs
D(Φ(f )) ≥ Cd(x)s (f (x)Φ′(f (x))− Φ(f (x)))
holds pointwise in Ω.This generalizes the Cordoba-Cordoba inequality from Rd
(d(x) =∞). The proof follows from approximation, convexity andthe fact that Θ = et∆1 obeys 0 ≤ Θ ≤ 1 and
ΛsD1 =
∞0
t−1− s2 (1−Θ(x , t))dt ≥ cd(x)−s
Dramatically different from Rd !
![Page 116: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/116.jpg)
The nonlinear bound for derivatives
Theorem(C, Ignatova) Let f ∈ L∞(Ω) ∩ D(Λs
D), 0 ≤ s < 2. Assume that f = ∂qwith q ∈ L∞(Ω) and ∂ a first order derivative. Then there existconstants c, C depending on Ω and s such that
f ΛsDf − 1
2Λs
Df 2 ≥ c‖q‖−sL∞ |fd |
2+s
holds pointwise in Ω, with
|fd (x)| =
|f (x)| if |f (x)| ≥ C‖q‖L∞(Ω)
1d(x) ,
0 if |f (x)| ≤ C‖q‖L∞(Ω)1
d(x) ,
Proof: nontrivial, uses precise bounds on the heat kernel and
f ΛsDf − 1
2Λs
Df 2 ≥ cs
2
∞0
t−1− s2 dt
Ω
HD(t , x , y)(f (x)− f (y))2dy
![Page 117: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/117.jpg)
The nonlinear bound for derivatives
Theorem(C, Ignatova) Let f ∈ L∞(Ω) ∩ D(Λs
D), 0 ≤ s < 2. Assume that f = ∂qwith q ∈ L∞(Ω) and ∂ a first order derivative. Then there existconstants c, C depending on Ω and s such that
f ΛsDf − 1
2Λs
Df 2 ≥ c‖q‖−sL∞ |fd |
2+s
holds pointwise in Ω, with
|fd (x)| =
|f (x)| if |f (x)| ≥ C‖q‖L∞(Ω)
1d(x) ,
0 if |f (x)| ≤ C‖q‖L∞(Ω)1
d(x) ,
Proof: nontrivial, uses precise bounds on the heat kernel and
f ΛsDf − 1
2Λs
Df 2 ≥ cs
2
∞0
t−1− s2 dt
Ω
HD(t , x , y)(f (x)− f (y))2dy
![Page 118: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/118.jpg)
Good cutoff
LemmaLet Ω be a bounded domain with C2 boundary. For ` > 0 smallenough (depending on Ω) there exist cutoff functions χ with theproperties: 0 ≤ χ ≤ 1, χ(y) = 0 if d(y) ≤ `
4 , χ(y) = 1 for d(y) ≥ `2 ,
|∇kχ| ≤ C`−k with C independent of ` and
Ω
(1− χ(y))
|x − y |d+j dy ≤ C1
d(x)j
and Ω
|∇χ(y)| 1|x − y |d
≤ C1
d(x)
hold for j ≥ 0 and d(x) ≥ `. We will refer to such χ as a “good cutoff”.
![Page 119: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/119.jpg)
Nonlinear bound, finite differencesTheoremLet Ω be a bounded domain with smooth boundary. There exists aconstant C such that, for every f ∈ C∞0 (Ω)
D(f ) = f ΛDf − 12
ΛDf 2 ≥ Cd(x)
f 2(x)
holds for all x ∈ Ω. Let χ ∈ C∞0 (Ω) be a good cutoff with scale ` > 0and let
f (x) = χ(x)(δhq(x)) = χ(x)(q(x + h)− q(x)).
Then
(f ΛDf )(x)− 12
(ΛDf 2)(x) ≥ γ1|h|−1 |fd (x)|3
‖q‖L∞+ γ1
f 2(x)
d(x)
holds pointwise in Ω when |h| ≤ `16 , and d(x) ≥ ` with
|fd (x)| = |f (x)|, if |f (x)| ≥ M‖q‖L∞(Ω)|h|
d(x).
![Page 120: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/120.jpg)
Commutator
Let χ be a good cutoff.
LemmaThere exists a constant Γ0 such that the commutator
Ch(θ) = χδhΛDθ − ΛD(χδhθ)
obeys
|Ch(θ)(x)| ≤ Γ0|h|
d(x)2 ‖θ‖L∞(Ω)
for d(x) ≥ `, |h| ≤ `16 .
![Page 121: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/121.jpg)
Finite difference of Riesz transform
LemmaLet χ be a good cutoff, and let u be defined by
u = R⊥D θ.
Then
|δhu(x)| ≤ C(√
ρD(f )(x) + ‖θ‖L∞
(|h|
d(x)+|h|ρ
)+ |δhθ(x)|
)holds for d(x) ≥ `, ρ ≤ cd(x), f = χδhθ and with C a constant
depending on Ω.
This gives a bound on |h|−1|δhu(x)| which costs D(f ).
![Page 122: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/122.jpg)
Finite difference of Riesz transform
LemmaLet χ be a good cutoff, and let u be defined by
u = R⊥D θ.
Then
|δhu(x)| ≤ C(√
ρD(f )(x) + ‖θ‖L∞
(|h|
d(x)+|h|ρ
)+ |δhθ(x)|
)holds for d(x) ≥ `, ρ ≤ cd(x), f = χδhθ and with C a constant
depending on Ω.This gives a bound on |h|−1|δhu(x)| which costs D(f ).
![Page 123: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/123.jpg)
Idea of proof of Holder bound
Good cutoff, and equation for δhθ imply:
12
Lχ (δhθ)2 + D(f ) + (δhθ)Ch(θ) = 0
withLχg = ∂tg + u · ∇xg + δhu · ∇hg + ΛD(χ2g).
andD(f ) ≥ γ1|h|−1‖θ‖−1
L∞ |(δhθ)d |3 + γ1(d(x))−1|δhθ|2
Multiply by |h|−2α with ε = α‖θ0‖L∞ small. Obtain:
Lχ
(δhθ(x)2
|h|2α
)+
γ1
4d(x)
(δhθ(x)2
|h|2α− Γ1`
−2α‖θ‖2L∞
)≤ 0.
![Page 124: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/124.jpg)
Idea of proof of Holder bound
Good cutoff, and equation for δhθ imply:
12
Lχ (δhθ)2 + D(f ) + (δhθ)Ch(θ) = 0
withLχg = ∂tg + u · ∇xg + δhu · ∇hg + ΛD(χ2g).
andD(f ) ≥ γ1|h|−1‖θ‖−1
L∞ |(δhθ)d |3 + γ1(d(x))−1|δhθ|2
Multiply by |h|−2α with ε = α‖θ0‖L∞ small.
Obtain:
Lχ
(δhθ(x)2
|h|2α
)+
γ1
4d(x)
(δhθ(x)2
|h|2α− Γ1`
−2α‖θ‖2L∞
)≤ 0.
![Page 125: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/125.jpg)
Idea of proof of Holder bound
Good cutoff, and equation for δhθ imply:
12
Lχ (δhθ)2 + D(f ) + (δhθ)Ch(θ) = 0
withLχg = ∂tg + u · ∇xg + δhu · ∇hg + ΛD(χ2g).
andD(f ) ≥ γ1|h|−1‖θ‖−1
L∞ |(δhθ)d |3 + γ1(d(x))−1|δhθ|2
Multiply by |h|−2α with ε = α‖θ0‖L∞ small. Obtain:
Lχ
(δhθ(x)2
|h|2α
)+
γ1
4d(x)
(δhθ(x)2
|h|2α− Γ1`
−2α‖θ‖2L∞
)≤ 0.
![Page 126: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/126.jpg)
Outlook
I Electroconvection: different configurations
I Electroconvection: analogy with Rayleigh-Benard, bounds onNusselt number.
I Electroconvection: complex fluids modelsI Electroconvection and SQG: free boundaries.
![Page 127: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/127.jpg)
Outlook
I Electroconvection: different configurationsI Electroconvection: analogy with Rayleigh-Benard, bounds on
Nusselt number.
I Electroconvection: complex fluids modelsI Electroconvection and SQG: free boundaries.
![Page 128: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/128.jpg)
Outlook
I Electroconvection: different configurationsI Electroconvection: analogy with Rayleigh-Benard, bounds on
Nusselt number.I Electroconvection: complex fluids models
I Electroconvection and SQG: free boundaries.
![Page 129: Nonlocal evolution equations, SQG · Collaborators I Tarek Elgindi (Princeton) I Mihaela Ignatova, (Princeton) I Vlad Vicol (Princeton)](https://reader031.vdocuments.mx/reader031/viewer/2022041515/5e2ae9fdf672d8160566ad8b/html5/thumbnails/129.jpg)
Outlook
I Electroconvection: different configurationsI Electroconvection: analogy with Rayleigh-Benard, bounds on
Nusselt number.I Electroconvection: complex fluids modelsI Electroconvection and SQG: free boundaries.